Thinking like...

So I’m focusing heavily on math these days, learning the content, working with math educators, and talking with mathematicians. The inevitable question arises: “Why do kids need to learn this stuff?” I’m not talking about arithmetic -- adding, subtracting, multiplying, and dividing whole numbers. We count stuff all the time. We keep track of batting averages; how much money we owe, spend, and save; how much more we need; how much extra (hopefully!) we have; what it means to double or halve a recipe; how many zombies we need to shoot, points to score, or crystals to capture to get to the next video game level. Arithmetic is part of our lives. But what about the math that comes after arithmetic? Why do we need to learn algebra? How often do we solve equations with exponents in daily life? It often feels that the reason we learn math beyond arithmetic is to do more math in school. No wonder kids find it boring and struggle to see the value.

When I talk to mathematicians, they don’t get it. Math is so exciting; it’s arithmetic that’s kind of boring. In fact, it’s a bit embarrassing when I describe FASTT Math to a mathematician, and he or she confides that she’s not very good with her math facts. How can an advanced mathematician not be good with math facts? Well, advanced math often doesn’t have much arithmetic; it doesn’t even have many numbers. What’s the deal?

The apparent conundrum got me thinking about my own training and teaching. In college I learned how to be a historian, and I learned how to teach history. They’re not the same. In fact, I never really liked history very much, but I loved being a historian. The facts of history -- the dates of events, the order of Presidents, the names and places of battles -- held little lasting interest. I haven’t used them, and, not surprisingly, I’ve forgotten many of them. However, what I learned doing history, being a historian, I use everyday. Doing history means deciphering the truth through the lenses of whatever evidence is available. What caused the Civil War? Why did Truman drop the atomic bomb? How did African tribal leaders feel about European explorers? So many witnesses and viewpoints to sort through, understand, and weigh. Lots of people seeing the same event and describing it in different ways. What really happened?

Anyone who has children exercises this way of thinking like a historian all the time. He did it. No she did. I use these skills as a husband and father, a teacher, and a manager. I collect and weigh evidence. I consider the biases of the sources and compare their versions with other available objective and subjective evidence. The skill of thinking like a historian has long out-lived the content of the history I learned. But that’s okay because the content was the vehicle to hone these skills. It served its purpose, and I know how to find it if and when I should need it again.

These historical thinking skills unfortunately frequently got lost in the obligation I felt as a teacher to convey the content. Indeed, the historical information became the end, it’s retention the measure of educational success. How do you measure thinking like a historian anyway?

I get the sense that we’ve followed a similar path in math. We don’t spend enough time helping kids see that learning the content of math is a path to a way of thinking and problem solving that they can use for the rest of their lives, long after they’ve forgotten a particular formula. You can read about mathematical thinking and rigor, but it’s tough to find in the standards. Ask an adult when they think mathematically, and they’ll typically describe an exercise dealing with numbers or spreadsheets.

But thinking mathematically thrives in the non-numeric world. The concepts of equivalence, commutativity, and order, for instance, have implications in many aspects of our everyday lives. Can I combine these recipe ingredients in any order or will the results differ if I add the egg first? Is there another way for me to get the outcome I want? How can I break down this complicated situation into more manageable pieces?

The teacher training books from Singapore, a nation that produces the highest performing math students in the world, talk specifically about the goal of mathematical thinking. But even they acknowledge the difficulty of focusing on this nebulous outcome compared to concreteness of specific math skills. Without our help students won’t see the power of thinking like a mathematician any more than they see the daily value of thinking like a historian. We’ve got to make the connections explicit and show how the exercise of learning the math (or history) makes those skills stronger. That’s a good design challenge. I’ll keep you posted.